In this follow-up to our post on basic concepts, we will explore the dynamics generated by models of linear vibrations.
Linear models of vibrating structures have the form
\[\begin{equation} M\ddot{x} + C\dot{x} + Kx = F, \label{eq:basic_vibrations} \end{equation}\]where $x(t) \in \R^N$ is a vector function representing the displacement from equilibrium, $F$ are external forces, $M$ is the mass, $C$ is the damping, and $K$ is the stiffness.
The differential equation \eqref{eq:basic_vibrations} can be rewritten as the first order ODE
\[\begin{align} \frac{d}{dt} \begin{bmatrix} x \\ \dot{x} \end{bmatrix} &= \Omega \begin{bmatrix} x \\ \dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ M^{-1}F \end{bmatrix} \notag\\ &:= \begin{bmatrix} 0 & I \\ -M^{-1}K & -M^{-1}C \end{bmatrix} \begin{bmatrix} x \\ \dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ M^{-1}F \end{bmatrix}; \label{eq:rewriting_main} \end{align}\]recall that the symbol ($:=$) is used to introduce a definition, in this case for $\Omega$. The general solution of this equation is
\[\begin{equation*} \begin{bmatrix} x(t) \\ \dot{x}(t) \end{bmatrix} = U(t) \begin{bmatrix} x_0 \\ \dot{x}_0 \end{bmatrix} + \int_0^t U(t - \tau) \begin{bmatrix} 0 \\ M^{-1}F(\tau) \end{bmatrix} \,d\tau, \end{equation*}\]where $x_0$ is the initial position, $\dot{x}_0$ is the initial velocity, and $U$ is the solution operator of the homogeneous equation ($F = 0$). If $(u, v)^t$ is a column vector, then $U(t)(u, v)^t$ is a solution of the homogeneous system with initial conditions $x_0 = u$ and $\dot{x}_0 = v$.
As we saw in our previous post, if $\Phi := (\phi_1, \ldots, \phi_N)$ is the basis of mass-normalized mode shapes, $Z := \diag(z_1, \ldots, z_N)$ the eigenvalues, and
\[\begin{equation*} L := \begin{bmatrix} \Phi & \bar{\Phi} \\ \Phi Z & \bar{\Phi}\bar{Z} \end{bmatrix}, \end{equation*}\]then
\[\begin{equation*} L^{-1}\Omega L = \begin{bmatrix} Z & \\ & \bar{Z} \end{bmatrix}. \end{equation*}\]Therefore, the solution operator is
\[\begin{align*} U(t) &= L \begin{bmatrix} e^{Zt} & \\ & e^{\bar{Z}t} \end{bmatrix} L^{-1} \\ &= \begin{bmatrix} -\im\Big(\Phi \frac{e^{Zt}}{Z\im(Z)}\Phi^t\Big)K & \im\Big(\Phi \frac{e^{Zt}}{\im(Z)}\Phi^t\Big)M \\ -\im\Big(\Phi \frac{e^{Zt}}{\im(Z)}\Phi^t\Big)K & \im\Big(\Phi \frac{Ze^{Zt}}{\im(Z)}\Phi^t\Big)M \end{bmatrix}. \end{align*}\]When the damping is proportional, the mode shapes are real and we can take them outside the $\im()$ operation.
During modal testing, the piece is initially at rest ($x_0 = \dot{x}_0 = 0$), and the force has the form $F(t) = wf(t)$, for $w \in \R^N$ and $f$ a scalar function, so the evolution of the system is
\[\begin{equation} x(t) = \im\Big[\Phi \frac{1}{\im(Z)}\Big(\int_0^t e^{Z(t - \tau)}f(\tau)\,d\tau\Big) \Phi^t w\Big]. \label{eq:evolution} \end{equation}\]If we extend the force to the past as $f = 0$ so that $f$ is a function in $\R$, then the components of the integral between parentheses equals the convolution $(1_{\R _+}e^{z_kt}) \star f$, where $1 _{\R _+}$ is the indicator function
\[\begin{equation*} 1_{\R _+}(t) = \begin{cases} 1 & \text{if } t > 0, \\ 0 & \text{otherwise.} \end{cases} \end{equation*}\]Taking the Fourier transform of \eqref{eq:evolution}, and assuming that $\re(Z) < 0$, we get
\[\begin{gather*} \hat{x}(\omega) = (G(\omega) + \bar{G}(-\omega))w\hat{f}(\omega) \\ G(\omega) := -\frac{1}{4\pi} \Phi \frac{1}{\im(Z)}\Big(\frac{1}{\omega + iZ/(2\pi)}\Big) \Phi^t, \end{gather*}\]or component-wise
\[\begin{equation*} G_{lm}(\omega) = -\frac{1}{4\pi}\sum_{k=1}^N \frac{\phi _{k,l}\phi _{k,m}}{\im(z_k)(\omega + iz_k/(2\pi))}, \end{equation*}\]where $\phi_k = (\phi_{k, 1}, \cdots, \phi_{k, N})$ are the mode shapes.
If the system is initially at rest (and not otherwise), then
\[\begin{equation*} (-(4\pi^2)\omega^2 M + 2\pi i \omega C + K)\hat{x}(\omega) = w\hat{f}(\omega). \end{equation*}\]Hence, we conclude that the transfer function is
\[\begin{equation*} (-(4\pi^2)\omega^2 M + 2\pi i \omega C + K)^{-1} = G(\omega) + \bar{G}(-\omega). \end{equation*}\]These relationships are important to interpret the results of modal tests.